Applied Probability And Stochastic Processes 2nd Edition Frank Beichelt - Solutions

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Show that the only.continuous solution of the functional equation g(s) = g(s)g(t) is g(s) = cs.
If X and Y are independent exponential random variables with respective means 1/A, and 1/A, compute the distribution of Z = min(X, Y) What is the conditional distribution of Z given that Z = X
Let Y, Y,. . be independent and identically distributed with P{Y = 0} = P{Y,y} (1a)e, y >0.Define the random variables X, n 0 by X = 0 = Xn+ax +Y+1 Prove that P{X = 0} = " P{X>x} (1 a")e, x>0
For a nonnegative random variable X, show that for a > 0, P{Xa} E[X]/a' Then use this result to show that n' (n/e)"
Consider an 7-sided coin and suppose that on each flip one of the sides appears side i with probability P., numbers r,,n,, let N, denote the number of flips required until side i has appeared for the n, time, i = 1,.,r, and let N = min N, P. 1 For given Thus N is the number of flips required until
Consider an elevator that starts in the basement and travels upward. Let N, denote the number of people that get in the elevator at floor i. Assume the N, are independent and that N, is Poisson with mean A, Each person entering at i will, independent of everything else, get off at j with
Suppose that shocks occur according to a Poisson process with rate A, and suppose that each shock, independently, causes the system to fail with probability p Let N denote the number of shocks that it takes for the system to fail and let 7 denote the time of failure Find P{N = n|T = t}
Events, occurring according to a Poisson process with rate A, are regis- tered by a counter. However, each time an event is registered the counter becomes inoperative for the next b units of time and does not register any new events that might occur during that interval Let R(t) denote the number
Cars pass a certain street location according to a Poisson process with rate A person wanting to cross the street at that location waits until she can see that no cars will come by in the next 7 time units. Find the expected time that the person waits before starting to cross (Note, for instance,
Buses arrive at a certain stop according to a Poisson process with rate A. If you take the bus from that stop then it takes a time R, measured from the time at which you enter the bus, to arrive home. If you walk from the bus stop then it takes a time W to arrive home. Suppose that your policy when
Suppose that events occur according to a Poisson process with rate A. Each time an event occurs we must decide whether or not to stop, with our objective being to stop at the last event to occur prior to some specified time T. That is, if an event occurs at time t, 0 t T and we decide to stop, then
Generating a Poisson Random Variable. Let U, U,... be independent uniform (0, 1) random variables. (a) If X, (-log U.)/A, show that X, is exponentially distributed with rate A. (b) Use part (a) to show that N is Poisson distributed with mean A when N is defined to equal that value of n such that +1
Compute the joint distribution of S1, S2, S3.
A machine needs two types of components in order to function. We have a stockpile of n type-1 components and m type-2 components.Type-i components last for an exponential time with rate , before failing. Compute the mean length of time the machine is operative if a failed component is replaced by
Show that Definition 2.1.1 of a Poisson process implies Definition 2.1.2.
In Example 16(A) if server i serves at an exponential rate A,, i = 1, 2, compute the probability that Mr. A is the last one out
If X, X2,...,X, are independent and identically distributed exponential random variables with parameter A, show that " X, has a gamma distri- bution with parameters (n, ) That is, show that the density function of X, is given by = f(t) e(t)/(n - 1)!, t0
Verify the formulas given for the mean and variance of an exponential random variable
If P{0 X a} = 1, show that Var(X) a/4.
If X is a nonnegative integer-valued random variable then the function P(z), defined for |z| 1 by 00 P(z) = E{z^]=ZzP{X=j}, 1-0 is called the probability generating function of X (a) Show that dk dzk P(z) = = k!P{X = k}. (b) With 0 being considered even, show that P{X is even} P(-1)+P(1) 2 (c) If X
Consider a round-robin tournament having n contestants, and let k, k it is possible for the tournament outcome to be such that for every set of k contestants there is a contestant who beat every member of this set.
A round-robin tournament of n contestants is one in which each of the (2) pairs of contestants plays each other exactly once, with the outcome of any play being that one of the contestants wins and the other loses Suppose the players are initially numbered 1, 2,.,n. The permutation ,, i, is called
Let X and X2 be independent Poisson random variables with means A and A. (a) Find the distribution of X, + X (b) Compute the conditional distribution of X, given that X + X = n
Let X denote the number of white balls selected when k balls are chosen at random from an urn containing n white and m black balls. Compute E[X] and Var(X).
Let X1, X2, be independent and identically distributed continuous random variables We say that a record occurs at time n, n > 0 and has value X, if X, > max(X, X-1), where x = -0. (a) Let N, denote the total number of records that have occurred up to (and including) time n. Compute E[N] and Var(N).
Suppose that n independent trials-each of which results in either out- come 1, 2, .., r with respective probabilities P1, P2, p,--are per- formed, p. 1 Let N, denote the number of trials resulting in outcome i(a) Compute the joint distribution of N, ..., N, This is called the multinomial
Compute the mean and variance of a binomial random variable with parameters n and p
Let X, denote a binomial random variable with parameters (n, p,), n 1 If np, as no, show that P{X} e^/i' as n o.
If X is a continuous random variable having distribution F show that. (a) F(X) is uniformly distributed over (0, 1), (b) if U is a uniform (0, 1) random variable, then F(U) has distribution F, where F(x) is that value of y such that F(y) = x
Consider the following method of shuffling a deck of n playing cards, numbered 1 through n. Take the top card from the deck and then replace it so that it is equally likely to be put under exactly k cards, for k = 0, 1,.,n 1. Continue doing this operation until the card that was initially on the
A fair die is continually rolled until an even number has appeared on 10 distinct rolls. Let X, denote the number of rolls that land on side i. Determine (a) E[X]. (b) E[X2]. (c) the probability mass function of X (d) the probability mass function of X2.
Consider a gambler who wins or loses 1 unit on each play with respective possibilities p and 1 - p. What is the probability that, starting with n units, the gambler will play exactly n + 2i games before going broke? (Hint Make use of ballot theorem.)
In the ballot problem compute the probability that A is never behind in the count of the votes.
Consider a gambler who on each gamble is equally likely to either win or lose 1 unit Starting with i show that the expected time until the gambler's fortune is either 0 or k is i(ki), i = 0,..., k. (Hint Let M, denote this expected time and condition on the result of the first gamble)
In Problem 123, let E[T] denote the expected time until the particle reaches 1 (a) Show that 1/(2p-1) if p > 1/2 E[T] = 00 if p 1/2 (b) Show that, for p > 1/2, 4p(1-p) Var(T) = (2p-1) (c) Find the expected time until the particle reaches n, n > 0. (d) Find the variance of the time at which the
Consider a particle that moves along the set of integers in the following manner If it is presently at i then it next moves to i + 1 with probability p and to i - 1 with probability 1 p Starting at 0, let a denote the probability that it ever reaches 1. (a) Argue that a p+(1 p)a. (b) Show that if p
The conditional variance of X, given Y, is defined by Var(XY) E[(X - E[XY])|Y] Prove the conditional variance formula, namely, Var(X) E[Var(X|Y)] + Var(E[X|Y]). Use this to obtain Var(X) in Example 1 5(B) and check your result by differentiating the generating function
Let U, U, M(x) = M(y) dy + 1, x1 x> 1 be independent uniform (0, 1) random variables, and let N denote the smallest value of n, n 0, such that n+1 0 where U, 1 Show that N is a Poisson random variable with mean A (Hint. Show by induction on n, conditioning on U, that P{N = n} = e-^\"/n!)
A Continuous Random Packing Problem Consider the interval (0, x) and suppose that we pack in this interval random unit intervalswhose left-hand points are all uniformly distributed over (0, x-1)-as follows. Let the first such random interval be I If I.... I have already been packed in the interval,
Let X, ., X, be independent and identically distributed continuous random variables having distribution F. Let X,,, denote the ith smallest of X, X, and let F., be its distribution function. Show that(a) F(x) = F(x) F-1-1(x) + F(x) F-1(x) === n-i (b) Fin-(x) = F(x) +- Fin(x) n n (Hints: For part
Let f(x) and g(x) be probability density functions, and suppose that for some constantc, f(x) = cg(x) for all x. Suppose we can generate random variables having density function g, and consider the following algorithm. Step 1: Generate Y, a random variable having density function g. Step 2:
Let F be a continuous distribution function and let U be a uniform (0, 1) random variable (a) If XF-(U), show that X has distribution function F. (b) Show that log(U) is an exponential random variable with mean 1.
Let N denote a nonnegative integer-valued random variable. Show that E[N] = P{N k} = P{N > k}. k=1 k=0 In general show that if X is nonnegative with distribution F, then E[X] = F(x) dx and Snx^-F(x) dx. E{X"] = snx"
Let X1, X2, ., X, be independent continuous random variables with common density function f Let X,, denote the ith smallest of X, .., X. (a) Note that in order for X,, to equal x, exactly i - 1 of the X's must be less than x, one must equal x, and the other ni must all be greater than x Using this
Consider a regenerative process satisfying the conditions of Theorem 3.7 1. Suppose that a reward at rate r(j) is earned whenever the process is in state j. If the expected reward during a cycle is finite, show that the long-run average reward per unit time is, with probability 1, given by r(X(s))
Consider successive flips of a fair coin. (a) Compute the mean number of flips until the pattern HHTHHTT ap- pears. (b) Which pattern requires a larger expected time to occur: HHTT or HTHT?
Prove Equation (3.5.3).
In Problem 3.9 suppose that potential customers arrive in accordance with a renewal process having interarrival distribution F. Would the number of events by time t constitute a (possibly delayed) renewal process if an event corresponds to a customer: (a) entering the bank? (b) leaving the bank?
Let A(t) and Y(t) denote respectively the age and excess at t. Find: (a) P{Y(t)>x| A(t) = s}. (b) P{Y(t)>x|A(t + x/2) = s}. (c) P{Y(t)>x|A(t + x) > s} for a Poisson process. (d) P{Y(t) > x, A(t) > y}. (e) If is called a renewal-type equation. In convolution notation the above states that g=h+g* F.
Let A(r) and Y(t) denote the age and excess at t of a renewal-process Fill in the missing terms: _? (a) A(t)>x0 events in the interval. (b) Y(t) > x0 events in the interval ? (c) P{Y(t) > x} = P{A( )> }. (d) Compute the joint distribution of A(t) and Y(r) for a Poisson process
A process is in one of n states, 1, 2, .,n Initially it is in state 1, where it remains for an amount of time having distribution F. After leaving state 1 it goes to state 2, where it remains for a time having distribution F2. When it leaves 2 it goes to state 3, and so on. From staten it returns
Show how Blackwell's theorem follows from the key renewal theorem.
Consider a miner trapped in a room that contains three doors. Door 1 leads her to freedom after two-days' travel; door 2 returns her to her room after four-days' journey, and door 3 returns her to her room after eight-days' journey. Suppose at all times she is equally to choose any of the three
Let X, X2,.. be independent and identically distributed with E[X]derive another expression for the limit in part (a). (c) Equate the two expressions to obtain Wald's equation
The random variables X,, X, are said to be exchangeable if X,, ., X, has the same joint distribution as X,., X, whenever i, i, , i, is a permutation of 1, 2, , n That is, they are exchangeable if the joint distribution function P{X, x, X x2, ..., X, x} is a symmetric function of (x, x2, .,x,). Let
If F is the uniform (0, 1) distribution function show that m(t) e 1, 0 i1 Now argue that the expected number of uniform (0, 1) random variables that need to be added until their sum exceeds 1 has mean e
On each bet a gambler, independently of the past, either wins or loses 1 unit with respective probabilities p and 1 -p Suppose the gambler's strategy is to quit playing the first time she wins k consecutive bets. At the moment she quits (a) find her expected winnings (b) find the expected number of
Consider successive flips of a coin having probability p of landing heads. Find the expected number of flips until the following sequences appear. (a) A (b) B = HHTTHH HTHTT. Suppose now that p = 1/2. (c) Find P{A occurs before B} (d) Find the expected number of flips until either A or B occurs
Packages arrive at a mailing depot in accordance with a Poisson process having rate A. Trucks, picking up all waiting packages, arrive in accor- dance to a renewal process with nonlattice interarrival distribution F Let X(t) denote the number of packages waiting to be picked up at time t. (a) What
In a k server queueing model with renewal arrivals show by counterexam- ple that the condition E[Y] < KE[X], where Y is a service time and X an interarrival time, is not sufficient for a cycle time to be necessarily finite. (Hint: Give an example where the system is never empty after the initial
For the queueing system of Section 3.61 define V(t), the work in the system at time, as the sum of the remaining service times of all customers in the system at Let V = lim V(s) ds/t. 00+ Also, let D, denote the amount of time the ith customer spends waiting in queue and define W = lim (D, + +
A system consisting of four components is said to work whenever both at least one of components 1 and 2 work and at least one of components 3 and 4 work Suppose that component i alternates between working and being failed in accordance with a nonlattice alternating renewal process with
Suppose in Example 33(A) that a coin's probability of landing heads is a beta random variable with parameters n and m, that is, the probability density is = f(p) Cp" (1p)-1, 0 p 1. Consider the policy that flips each newly chosen coin until m consecutive flips land tails, then discards that coin
The life of a car is a random variable with distribution F An individual has a policy of trading in his car either when it fails or reaches the age of A. Let R(A) denote the resale value of an A-year-old car. There is no resale value of a failed car Let C, denote the cost of a new car and suppose
For a renewal reward process show that lim E[RN+] E[RX] E[X] Assume the distribution of X, is nonlattice and that any relevant function is directly Riemann integrable. When the cycle reward is defined to equal the cycle length, the above yields lim E[XN+1] = 1++00 E[X2] E[X]' which is always
Prove Blackwell's theorem for renewal reward processes That is, assum- ing that the cycle distribution is not lattice, show that, as , E[reward in (1,a)] E[reward in cycle] E[time of cycle] Assume that any relevant function is directly Riemann integrable.
Consider a delayed renewal process {N(t), t 0} whose first interarrival has distribution G and the others have distribution F Let m(1): E[ND(t)] (a) Prove that m(t) = G(t) + fm(1 x) dG(x), - where m(t) == Fn(t) (b) Let A,(t) denote the age at time Show that if F is nonlattice with fx dF(x) and
Draw cards one at a time, with replacement, from a standard deck of playing cards Find the expected number of draws until four successive cards of the same suit appear.
A coin having probability p of landing heads is flipped k times Additional flips of the coin are then made until the pattern of the first k is repeated (possibly by using some of the first k flips). Show that the expected number of additional flips after the initial k is 2*.
Let {N(t), t 0} be a renewal process and suppose that for all n and t conditional on the event that N(t) = n, the event times S,., S, are distributed as the order statistics of a set of independent uniform (0, t) random variables Show that {N(t), t 0} is a Poisson process (Hint Consider E[N(s) |
Prove that the renewal function m(t), 0 t < uniquely determines the interarrival distribution F
Prove the renewal equation m(t) = F(t) + f' m(t x) dF(x). -
Let T1, T2, .. denote the interarrival times of events of a nonhomoge- neous Poisson process having intensity function (t) (a) Are the 7, independent? (b) Are the 7, identically distributed" (c) Find the distribution of T (d) Find the distribution of T
Complete the proof that for a nonhomogeneous Poisson process N(t + s) N(t) is Poisson with mean m(t + s) - m(t)
Compute the moment generating function of D(t) in Example 23(C).
Compute the conditional distribution of S, S, ., S, given that S, = t
Suppose that cars enter a one-way highway of length L in accordance with a Poisson process with rate Each car travels at a constant speed that is randomly determined, independently from car to car, from the distribution F. When a faster car encounters a slower one, it passes it with no loss of time
For the model of Example 2.3(C), find (a) Var[D(t)]. (b) Cov[D(t), D(t + s)]
Suppose cars enter a one-way infinite highway at a Poisson rate A The ith car to enter chooses a velocity V, and travels at this velocity Assume that the V,'s are independent positive random variables having a common distribution F. Derive the distribution of the number of cars that are located in
Individuals enter the system in accordance with a Poisson process having rate Each arrival independently makes its way through the states of the system Let a,(s) denote the probability that an individual is in state i a time s after it arrived Let N,(t) denote the number of individuals in state i
Suppose that each event of a Poisson process with rate A is classified as being either of type 1, 2, .k If the event occurs at s, then, indepen- dently of all else, it is classified as type i with probability P,(s), i = 1, .., k, P(s) = 1. Let N,(t) denote the number of type i arrivals in [0, 1]
Suppose that each event of a Poisson process with rate A is classified as being either of type 1, 2, , k If the event occurs at s, then, indepen- dently of all else, it is classified as type i with probability P,(s), i = 1, .., k, P(s) = 1. Let N,(t) denote the number of type i arrivals in [0, 1]
Busloads of customers arrive at an infinite server queue at a Poisson rate A Let G denote the service distribution. A bus contains j customers with probability a,,j = 1,. Let X(t) denote the number of customers that have been served by time t (a) E[X(t)] = ? (b) Is X(r) Poisson distributed?
Consider a nonhomogeneous Poisson process {N(t), t = 0}, where A(t)>0 for all t. Let N*(t) = N(m(t)) Show that {N(t), t 0} is a Poisson process with rate = 1
(a) Let {N(t), t = 0} be a nonhomogeneous Poisson process with mean value function m(t) Given N(t) = n, show that the unordered set of arrival times has the same distribution as n independent and identically distributed random variables having distribution function F(x)= m(x) m(t) x t x>1 (b)
Is it true that (a) N(t) t? (b) N(t) n if and only if S, t? (c) N(t)>n if and only if S,
Consider a conditional Poisson process where the distribution of A is the gamma distribution with parameters m and a that is, the density is given by - g(A) = aea (a)/(m 1)', 0 < x < . (a) Show that m+n-1 P{N(t) = n}= n (m + n 1 ) (+) (+) * m n0. a+(b) Show that the conditional distribution of A
For a conditional Poisson process. (a) Explain why a conditional Poisson process has stationary but not independent increments. (b) Compute the conditional distribution of A given {N(s), 0 st}, the history of the process up to time t, and show that it depends on the history only through N(t).
Give an example of a counting process {N(t), 0} that is not a Poisson process but which has the property that conditional on N(t) = n the first n event times are distributed as the order statistics from a set of n independent uniform (0, t) random variables.
Compute Cov(X(s), X(t)) for a compound Poisson process
Let {X(t), t 0} be a compound Poisson process with X(t) = X,, and suppose that A1 and P{X, j} = j/10, j = 1, 2, 3, 4. Calculate P{X(4) = 20).
Let {X(t), t 0} be a compound Poisson process with X(t) = X,, and suppose that the X, can only assume a finite set of possible values. Argue that, for large, the distribution of X(t) is approximately normal N(1) 1-1
Let C denote the number of customers served in an M/G/1 busy pe- riod. Find (a) E[C]. (b) Var(C)1=1
Let {N(t), t 0} be a nonhomogeneous Poisson process with intensity function A(t), t 0 However, suppose one starts observing the process at a random time 7 having distribution function F. Let N*(t) = N(T + t) N(7) denote the number of events that occur in the first time units of observation (a) Does
Repeat Problem 2 25 when the events occur according to a nonhomoge- neous Poisson process with intensity function (t), t 0
A two-dimensional Poisson process is a process of events in the plane such that (i) for any region of area A, the number of events in A is Poisson distributed with mean AA, and (ii) the numbers of events in nonoverlapping regions are independent. Consider a fixed point, and let X denote the
Let U(1), U() denote the order statistics of a set of n uniform (0, 1) random variables Show that given U(m) = y, U(1), U(n-1) are distributed as the order statistics of a set of n-1 uniform (0, y) random variables
In Example 36(C) suppose that the renewal process of arrivals is a Poisson process with mean Let N* denote that value of N that mini- mizes the long-run average cost if a train leaves whenever there are N travelers waiting Another type of policy is to have a train depart every T time units Compute
For another approach to proving that Definition 2.1.2 implies Defini- tion 2.1.1. (a) Prove, using Definition 2.1.2, that Po(ts) Po(t) Po(s) (b) Use (a) to infer that the interarrival times X1, X2,... are independent exponential random variables with rate A. (c) Use (b) to show that N(t) is Poisson
Let \(Y_{0}, Y_{1}, \ldots\) be a sequence of independent random variables, which are identically distributed as \(N(0,1)\). Are the stochastic sequences \(\left\{X_{0}, X_{1}, \ldots\right\}\) with(1) \(X_{n}=\sum_{i=0}^{n} Y_{i}^{2}\)(2) \(X_{n}=\sum_{i=0}^{n} Y_{i}^{3}\)(3)

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Applied Probability And Stochastic Processes 2nd Edition Frank Beichelt - Solutions

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